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A packing of a graph G is a subset P of the vertex set of G such that the closed neighborhoods of any two distinct vertices of P do not intersect. We study graphs with a unique packing of the maximum cardinality. We present several general properties for such graphs. These properties are used to characterize the trees with a unique maximum packing. Two characterizations are presented where one of them is inductive based on five operations.

Graphs with unique maximum packing of closed neighborhoods